1. AP Calculus AB Chapter 2, Section 6 Related Rates 2013 - 2014

2. Related rates

3. Related Rates For example: If water is draining out of a conical tank, what are the variables that are changing with respect to time?

4. Conical Tank continued Use implicit differentiation for the volume of a cone and describe what each piece means.

5. Two Rates That Are Related

6. Ripples in a Pond A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?

7. Guidelines for Solving Related-Rate Problems 1.Identify all given quantities and quantities to be determined. Make a sketch and label the quantities. 2.Write an equation involving the variables. 3.Use the Chain Rule, implicitly differentiate both sides of the equation with respect to time t. 4.After completing step 3, substitute into the resulting equation all known values for the variables and their rates of change. Solve if necessary.

8. Examples of mathematical models involving rates of change Verbal StatementMathematical Model The velocity of a car after traveling for 1 hour is 50 miles per hour Water is being pumped into a swimming pool at a rate of 10 cubic meters per hour A gear is revolving at a rate of 25 revolutions per minute (1 revolution = 2π rad)

9. An Inflating Balloon Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet.

10. The Speed of an Airplane Tracked by Radar An airplane is flying on a flight path that will take it directly over a radar tracking station. If s is decreasing at a rate of 400 miles per hour when s = 10 miles, what is the speed of the plane?

11. A Changing Angle of Elevation

13. The Velocity of a Piston

15. 2.6 Homework Pg. 154 – 156, #’s: 3, 7, 15, 19, 27, 31, 43 7 problems